Question

The number of four letter words that can be lone
using the letters of the word RAMANA is
no..

Answer 1

Answer:

Suppose you have a 4 letter string composed of, say, 1 distinct and 3 identical letters

There would be 4!1!3! permutations, also expressible as a multinomial coefficient, (41,3)

Similarly, for 2 distinct, 2 identical, and 3 distinct, 1 identical,

it would be (42,2) and (43,1) respectively.

In the polynomial expression 4!(1+x/1!)3(1+x+x2/2!+x3/3!),

the 4! corresponds to the numerator, whatever the combination; the first term in () corresponds to choosing one or more from R,M,N; and the other term corresponds to choosing 1,2, or 3A′s

It will become evident why this approach works if we expand the first term in ( ), and compare serially with your case approach by just using the appropriate coefficients to get terms in x4

4!(1+3x+3x2+x3)(1+x+x2/2!+x3/3!)

To find the coefficient of x4, consider the three cases that produce x4

One from R,M,N,3A′s:4!⋅3⋅13!=12

Two from R,M,N,2A′s:4!⋅3⋅12!=36

Three from R,M,N,1A:4!⋅1⋅1=24

Coefficient of x4=12+36+24=72

We can now clearly see why the coefficient of x4 in the expression automatically gives all possible permutations of 4 letters